We will describe the Cohen-Macaulay part of the Ziegler spectrum of the D∞ plane singularity S and calculate the nilpotency index of the radical of the category of finitely generated Cohen-Macaulay S-modules. IntroductionA local commutative noetherian ring is said to be of countable (or discrete) Cohen-Macaulay representation type if it has only countably (infinitely) many non-isomorphic finitely generated maximal Cohen-Macaulay modules. For complete 1-dimensional surfaces the classification of [3] shows that there are essentially two possibilities:The model theory of Cohen-Macaulay R-modules was investigated in [15]. In particular the Cohen-Macaulay part of the Ziegler spectrum of R, ZCM R , was described, and the nilpotency index of the radical of the category of finitely generated Cohen-Macaulay modules was calculated. In this paper we will address similar questions for the ring S. By factoring out x 2 we obtain a natural surjection S → R, hence an embedding from ZCM R onto a closed subset of ZCM S . Our initial feeling was that the Cantor-Bendixson rank of points in the former space will jump to higher values via this embedding (see [10] or [16] for dealing with this effect). However this is not the case -we will show that the points preserve their ranks in the ambient space; and the Cantor-Bendixson rank of ZCM S equals 2.From point of view of properties we will investigate in this paper, the category of Cohen-Macaulay modules over S is a 'double' of the corresponding category of R-modules. For instance the Krull-Gabriel dimension of the definable category generated by finitely generated Cohen-Macaulay modules equals 2 for R and for S. Further, similarly to the A ∞ -case, one can glue from the Auslander-Reiten quiver of the category of finitely generated Cohen-Macaulay S-modules the Ringel quilt, which is the Möbius stripe.This gives a global geometric structure to the above category including objects, but also morphisms, -the part which is so often neglected. Using this realization we will show that the nilpotency index of the radical of this category equals ω · 2. Note that, by Schröer [21, Prop. 6.1], for finite dimensional algebras this index cannot be equal to a limit ordinal.

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